The backdoor adjustment [1]}, [2]} allows us to remove \(do(x)\)  in \(P(y|do(x))\)  for easier estimation. We rewrite \(P(y|do(x))\)  w.r.t. the confounder \(S\)  as:
\(\begin{aligned}P(y|do(x))&=\sum \limits _{S=s\in \lbrace 0,1\rbrace }P(y|x,S=s)P(S=s) \\&=\frac{P(x,y,S=1)}{P(x|S=1)}+\frac{P(x,y,S=0)}{P(x|S=0)}.\end{aligned}\) 
